This is also the number of arrangements of n pairs which are equivalent under the joint operation of sequence reversal and permutations of labels. Assume that the elements of n distinct pairs are labeled to show the pair of origin, e.g., [1 1], [2 2]. The number of distinguishable ways of arranging these elements falls as the conditions are made more general:

a(n) = A000680: element order is significant and the labels are distinguishable;

b(n) = A001147: element order is significant but labels are not distinguishable, i.e. all label permutations of a given sequence are equivalent;

c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;

Those classes that do not map to themselves form reciprocal pairs of classes under the joint operation and their number is r(n). Then c = b - r/2 = b - (b - d)/2 = (b+d)/2. A formula for r(n) is not available, but formulas are available for b(n) = A001147 and d(n) = A047974, allowing an explicit formula for this sequence.

c(n) is useful in extracting structure information without regard to pair ordering (see example). c(n) terms also appear in formulas related to binary operators, eg, the number of binary operators in a k-valued logic that are invertible in 1 operation.

For 3 pairs, the arrangement A = [112323] is the same as B = [212133] under the permutation of the labels [123] -> [312] plus reversal of the elements, or vice versa. The unique structure common to A and B is {1 intact pair + 2 interleaved pairs}, where the order is not significant (contrast A001147). (End)